3.560 \(\int \frac{3+2 x}{729-64 x^6} \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{972} \log \left (4 x^2+6 x+9\right )-\frac{1}{486} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}} \]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(162*Sqrt[3]) - Log[3 - 2*x]/486 + Log[9 + 6*x + 4*x^2]/972

________________________________________________________________________________________

Rubi [A]  time = 0.0481794, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1586, 2058, 618, 204, 628} \[ \frac{1}{972} \log \left (4 x^2+6 x+9\right )-\frac{1}{486} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/(729 - 64*x^6),x]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(162*Sqrt[3]) - Log[3 - 2*x]/486 + Log[9 + 6*x + 4*x^2]/972

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{3+2 x}{729-64 x^6} \, dx &=\int \frac{1}{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5} \, dx\\ &=\int \left (-\frac{1}{243 (-3+2 x)}+\frac{1}{54 \left (9-6 x+4 x^2\right )}+\frac{3+4 x}{486 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac{1}{486} \log (3-2 x)+\frac{1}{486} \int \frac{3+4 x}{9+6 x+4 x^2} \, dx+\frac{1}{54} \int \frac{1}{9-6 x+4 x^2} \, dx\\ &=-\frac{1}{486} \log (3-2 x)+\frac{1}{972} \log \left (9+6 x+4 x^2\right )-\frac{1}{27} \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}}-\frac{1}{486} \log (3-2 x)+\frac{1}{972} \log \left (9+6 x+4 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0134169, size = 46, normalized size = 0.92 \[ \frac{1}{972} \left (\log \left (4 x^2+6 x+9\right )-2 \log (3-2 x)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x)/(729 - 64*x^6),x]

[Out]

(2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 2*Log[3 - 2*x] + Log[9 + 6*x + 4*x^2])/972

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 39, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( -3+2\,x \right ) }{486}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{972}}+{\frac{\sqrt{3}}{486}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+2*x)/(-64*x^6+729),x)

[Out]

-1/486*ln(-3+2*x)+1/972*ln(4*x^2+6*x+9)+1/486*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.3875, size = 51, normalized size = 1.02 \begin{align*} \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x^6+729),x, algorithm="maxima")

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*log(4*x^2 + 6*x + 9) - 1/486*log(2*x - 3)

________________________________________________________________________________________

Fricas [A]  time = 1.40525, size = 128, normalized size = 2.56 \begin{align*} \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x^6+729),x, algorithm="fricas")

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*log(4*x^2 + 6*x + 9) - 1/486*log(2*x - 3)

________________________________________________________________________________________

Sympy [A]  time = 0.16509, size = 46, normalized size = 0.92 \begin{align*} - \frac{\log{\left (x - \frac{3}{2} \right )}}{486} + \frac{\log{\left (4 x^{2} + 6 x + 9 \right )}}{972} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{486} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x**6+729),x)

[Out]

-log(x - 3/2)/486 + log(4*x**2 + 6*x + 9)/972 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/486

________________________________________________________________________________________

Giac [A]  time = 1.04905, size = 53, normalized size = 1.06 \begin{align*} \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-64*x^6+729),x, algorithm="giac")

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*log(4*x^2 + 6*x + 9) - 1/486*log(abs(2*x - 3))